This document discusses various sorting algorithms including merge sort. It begins with an introduction to sorting and searching. It then provides pseudocode for the merge sort algorithm which works by dividing the array into halves, recursively sorting the halves, and then merging the sorted halves back together. An example is provided to illustrate the merge sort process. Key steps include dividing, conquering by recursively sorting subarrays, and combining through merging. The overall time complexity of merge sort is O(n log n).

1. 14.1 IntroductIon to SEArcHInG
linear search binary search
14.2 LInEAr SEArcH
sequential search
A[]
LEArnInG objEctIvE
Searching and sorting are two of the most common operations in computer science.
Searching and
Sorting
chapter 14

2. Searching and Sorting 425
and the value to be searched is VAL = 7, then searching means
POS = 3
of POS and I while
be executed till I is less than N (total number of elements
VAL
the value of I
VAL
VAL and no match is found, then it means that VAL is not
Complexity of Linear Search Algorithm
Linear search executes in O(n) n
VAL
VAL is not
n
Programming ExamPlE
1.
#include <stdio.h>
#include <stdlib.h>
#include <conio.h>
}
}
}
LINEAR_SEARCH(A, N, VAL)
Step 1: [INITIALIZE] SET POS = -1
Step 2: [INITIALIZE] SET I = 1
Step 3: Repeat Step 4 while I<=N
Step 4: IF A[I] = VAL
SET POS = I
PRINT POS
Go to Step 6
[END OF IF]
[END OF LOOP]
Step 6: EXIT
SET I = I + 1
Step 5: IF POS = –1
PRINT VALUE IS NOT PRESENT
IN THE ARRAY
[END OF IF]
"
"
Figure 14.1 Algorithm for linear search

3. 426 Data Structures Using C
}
14.3 Binary Search
and the value to be searched is
and
is less than VAL
and
is less than VAL
and
and
and
is calculated as
and
POS
VAL is not equal to , then the values of , , and
VAL is smaller or greater than
(a) If , then VAL
(b) If , then VAL
VAL
, , and POS while
executed until is less than or equal to
is equal to VAL

4. Searching and Sorting 427
found, then the value of POS
is not found, and if the value of
is greater than VAL, the value of is
is greater
than VAL, then the value of
, then VAL is
Complexity of Binary Search Algorithm
,
n is the number of elements in the
n)
n
Programming ExamPlE
2.
##include <stdio.h>
#include <stdlib.h>
#include <conio.h>
}
}
BINARY_SEARCH(A, lower_bound, upper_bound, VAL)
Step 1: [INITIALIZE] SET BEG = lower_bound
END = upper_bound, POS = - 1
Step 2: Repeat Steps 3 and 4 while BEG <= END
Step 3: SET MID = (BEG + END)/2
Step 4: IF A[MID] = VAL
SET POS = MID
PRINT POS
Go to Step 6
ELSE IF A[MID] > VAL
SET END = MID - 1
ELSE
SET BEG = MID + 1
[END OF IF]
[END OF LOOP]
Step 5: IF POS = -1
PRINT “VALUE IS NOT PRESENT IN THE ARRAY”
[END OF IF]
Step 6: EXIT
Figure 14.2 Algorithm for binary search

5. 428 Data Structures Using C
else
}
}
}
}
}
}
}
14.4 inTerPOLaTiOn Search
calculation is done based on the values
INTERPOLATION_SEARCH (A, lower_bound, upper_bound, VAL)
Figure 14.3 Algorithm for interpolation search

6. Searching and Sorting 429
VAL
Low Item to be searched High
Middle = (low+ high)/2
(a) Binary search divides the list into two equal halves
VAL
Low Item to be searched High
Middle = low + (high - low) ( (key – a[low]) /(a[high] – a[low]))¥
(b) Interpolation search divides the list into two equal halves
Figure 14.4 Difference between binary search and interpolation search
Complexity of Interpolation Search Algorithm
When n
O(n)
example 14.1 Given a list of numbers . Search
Solution
Programming ExamPlE
3.
#include <stdio.h>
#include <conio.h>

7. 430 Data Structures Using C
else
}
}
else
}
14.5 JUMP Search
3 7
3 7
3 7

8. Searching and Sorting 431
JUMP_SEARCH (A, lower_bound, upper_bound, VAL, N)
POS = I
Figure 14.5
Advantage of Jump Search over Linear Search
Advantage of Jump Search over Binary Search
O(log n)
How to Choose the Step Length?
n
n
Further Optimization of Jump Search

9. 432 Data Structures Using C
Searching can start from somewhere middle in the list rather than from the beginning to optimize
performance.
n =
Complexity of Jump Search Algorithm
n
O
Programming ExamPlE
4.
#include <stdio.h>
#include <math.h>
#include <conio.h>
{
{
else
{
int main()
{

10. Searching and Sorting 433
else
Fibonacci Search
O(log n).
14.6 IntroductIon to SortInG
A A are
< A[1] < < ...... < A[N]
∑ Internal sorting
∑ External sorting
14.6.1 Sorting on Multiple Keys
∑

11. 434 Data Structures Using C
∑
∑
note sort key. A
primary sort key
the second is known as the secondary sort key
name department Salary Phone number
Janak 1000000 9812345678
Computer Science 890000 9910023456
900000 7838987654
Huma 1100000 9654123456
Computer Science 750000 9350123455
name department Salary Phone number
Computer Science 750000 9350123455
Computer Science 890000 9910023456
900000 7838987654
Huma 1100000 9654123456
Janak 1000000 9812345678
14.6.2 Practical considerations for Internal Sorting
∑
∑
∑
∑
∑
∑
14.7 bubbLE Sort

12. Searching and Sorting 435
bubble sorting
note
Technique
A[1] A[1]
A[1] A[1]
A[1] A[1]
A[1]
Example 14.2 A[]
Pass 1:
29, 52
63, 87
27, 87
19 87
54, 87
Pass 2:
29, 30
27, 63

13. 436 Data Structures Using C
19, 63
54, 63
Pass 3:
27, 52
, 19, 52
Pass 4:
27, 30
19, 30
Pass 5:
27, 29,
19, 29
Pass 6:
19, 27,
Pass 7:
(a) Com
N is
I is the
BUBBLE_SORT(A, N)
Step 1: Repeat Step 2 For 1 = to N-1
Step 2: Repeat For J = to N - I
Step 3: IF A[J] > A[J + 1]
SWAP A[J] and A[J+1]
[END OF INNER LOOP]
[END OF OUTER LOOP]
Step 4: EXIT
Figure 14.6

14. Searching and Sorting 437
Complexity of Bubble Sort
)
O(n )
n n
Programming ExamPlE
5. n
#include <stdio.h>
#include <conio.h>
int main()
{
{
{
{
{
1
Bubble Sort Optimization

15. 438 Data Structures Using C
{
{
{
{
Complexity of Optimized Bubble Sort Algorithm
O(n)
O(n )
14.8 InSErtIon Sort
sor
Technique
∑
∑
∑ n LB =
∑ LB =
∑

16. Searching and Sorting 439
Example 14.3
Solution
9 39 45 63 18 81 1 8 54 72 3639 9 45 63 18 81 1 8 54 72 36
A[ ] is the only element in sorted list (Pass 1)
9 39 45 63 18 81 1 8 54 72 369 39 45 63 18 81 1 8 54 72 36
(Pass 2) (Pass 3)
9 18 39 45 63 81 1 8 54 72 369 18 39 45 63 81 1 8 54 72 36
(Pass 4) (Pass 5)
9 18 39 45 54 63 81 1 8 72 369 18 39 45 63 81 1 8 54 72 36
(Pass 6) (Pass 7)
9 18 36 39 45 54 63 72 81 1 89 18 39 45 54 63 72 81 1 8 36
(Pass 8) (Pass 9)
39 9 45 63 18 81 1 8 54 72 36
Sorted Unsorted
is th A[1]
A
A[1] A[1]
A[K] A[1] ...
A[K] A[J]
A[J] <= A[K] A[K] in its correct
... A[J] A[K] is
th
Kth
Jth
Complexity of Insertion Sort
O(n)
O(n )
INSERTION-SORT (ARR, N)
Step 1: Repeat Steps 2 to 5 for K = 1 to N – 1
Step 2: SET TEMP = ARR[K]
Step 3: SET J = K - 1
Step 4: Repeat while TEMP <= ARR[J]
SET ARR[J + 1] = ARR[J]
SET J = J - 1
[END OF INNER LOOP]
Step 5: SET ARR[J + 1] = TEMP
[END OF LOOP]
Step 6: EXIT
Figure 14.7

17. 440 Data Structures Using C
Advantages of Insertion Sort
The advantages of this sorting algorithm are as follows:
∑
∑
∑
∑ i O(1)
∑ it is said t
Programming ExamPlE
6.
#include <stdio.h>
#include <conio.h>
void insertion_sort(int arr[], int n);
void main()
{
printf("n Enter the number of elements in the array: ");
scanf("%d", &n);
printf("n Enter the elements of the array: ");
for(i=0;i<n;i++)
{
scanf("%d", &arr[i]);
}
insertion_sort(arr, n);
printf("n The sorted array is: n");
for(i=0;i<n;i++)
printf(" %dt", arr[i]);
getch();
}
void insertion_sort(int arr[], int n)
{
int i, j, temp;
for(i=1;i<n;i++)
{
temp = arr[i];
j = i-1;
while((temp < arr[j]) && (j>=0))
{
arr[j+1] = arr[j];
j--;
}
arr[j+1] = temp;
}
}
Output
The sorted array is :
14.9 SELECTION SORT
gorit O(n )

18. Searching and Sorting 441
Technique
N
∑ POS
∑ POS
∑ POS
] so that ...
Example 14.4
PASS POS ARR[0] ARR[1] ARR[2] ARR[3] ARR[4] ARR[5] ARR[6] ARR[7]
1 1
Kth
POS ...
to
SELECTION SORT(ARR, N)
Step 1: Repeat Steps 2 and 3 for K = 1
to N-1
Step 2: CALL SMALLEST(ARR, K, N, POS)
Step 3: SWAP A[K] with ARR[POS]
[END OF LOOP]
Step 4: EXIT
SMALLEST (ARR, K, N, POS)
Step 1: [INITIALIZE] SET SMALL = ARR[K]
Step 2: [INITIALIZE] SET POS = K
Step 3: Repeat for J = K+1 to N
IF SMALL > ARR[J]
SET SMALL = ARR[J]
SET POS = J
[END OF IF]
[END OF LOOP]
Step 4: RETURN POS
-1
Figure 14.8
Complexity of Selection Sort
n

19. 442 Data Structures Using C
thus, n–1
n – 1
(n – 1) + (n – 2) + ... + 2 + 1
= n(n – 1) / 2 = O(n2
) comparisons
Advantages of Selection Sort
∑
∑
∑
Programming ExamPlE
7.
#include <stdio.h>
#include <stdlib.h>
#include <conio.h>
int smallest(int arr[], int k, int n);
void selection_sort(int arr[], int n);
void main(int argc, char *argv[]) {
int arr[10], i, n;
printf("n Enter the number of elements in the array: ");
scanf("%d", &n);
printf("n Enter the elements of the array: ");
for(i=0;i<n;i++)
{
scanf("%d", &arr[i]);
}
selection_sort(arr, n);
printf("n The sorted array is: n");
for(i=0;i<n;i++)
printf(" %dt", arr[i]);
}
int smallest(int arr[], int k, int n)
{
int pos = k, small=arr[k], i;
for(i=k+1;i<n;i++)
{
if(arr[i]< small)
{
small = arr[i];
pos = i;
}
}
return pos;
}
void selection_sort(int arr[],int n)
{
int k, pos, temp;
for(k=0;k<n;k++)
{
pos = smallest(arr, k, n);
temp = arr[k];

20. Searching and Sorting 443
arr[k] = arr[pos];
arr[pos] = temp;
}
}
14.10 MERGE SORT
Divide n n/2
A
A A1
and A2
A
Conquer
Combine n/2 n
∑
∑
∑
∑
∑
∑
Example 14.5
Solution
39 9 45 1881 279 72
(Divide and Conquer the array)
39 9 45 1881 279 72
182790 72
18279 72
18279 72
39 9 4581
39 9 4581
39 9 4581
(Combine the elements to form a sorted array)
9 18 39 927 7245 81
92718 729 39 8145
72927 189 39 8145
18279 7239 9 4581
merge
9 39 45 81 18 27 72
BEG, I MID J END
9
INDEX
TEMP
9

21. 444 Data Structures Using C
I or J
9 39 45 81 18 27 72
BEG MID J END INDEX
TEMP
I
9 189
9 39 45 81 18 27 72
BEG MID J INDEX
9
I
18 279
END
9 39 45 81 18 27 72
INDEX
9
BEG MID JI
18 27 399
END
9 39 45 81 18 27 72
INDEX
9
BEG MID JI
18 27 39 459
END
9 39 45 81 18 27 72
INDEX
9
BEG JI, MID
18 27 39 45 729
END
9 39 45 81 18 27 72
INDEX
9
BEG I, MID
18 27 39 45 72 819
J END
I is greater than
9 39 45 81 18 27 72
INDEX
9
BEG MID I
18 27 39 45 72 81 99
J END
MERGE (ARR, BEG, MID, END)
Step 1: [INITIALIZE] SET I = BEG, J = MID + 1, INDEX =
Step 2: Repeat while (I <= MID) AND (J<=END)
IF ARR[I] < ARR[J]
SET TEMP[INDEX] = ARR[I]
SET I = I + 1
ELSE
SET TEMP[INDEX] = ARR[J]
SET J = J + 1
[END OF IF]
SET INDEX = INDEX + 1
[END OF LOOP]
Step 3: [Copy the remaining elements of right sub-array, if any]
IF I > MID
Repeat while J <= END
SET TEMP[INDEX] = ARR[J]
SET INDEX = INDEX + 1, SET J = J + 1
[END OF LOOP]
[Copy the remaining elements of left sub-array, if any]
ELSE
Repeat while I <= MID
SET TEMP[INDEX] = ARR[I]
SET INDEX = INDEX + 1, SET I = I + 1
[END OF LOOP]
[END OF IF]
Step 4: [Copy the contents of TEMP back to ARR] SET K=
Step 5: Repeat while K < INDEX
SET ARR[K] = TEMP[K]
SET K = K + 1
[END OF LOOP]
Step 6: END

22. Searching and Sorting 445
Complexity of Merge Sort
The running time of merge sort in the average
case and the worst case can be given as O(n log
n). Although merge sort has an optimal time
complexity, it needs an additional space of O(n)
for the temporary array TEMP.
Programming ExamPlE
8. Write a program to implement merge sort.
#include <stdio.h>
#include <conio.h>
void merge(int a[], int, int, int);
void merge_sort(int a[],int, int);
void main()
{
printf("n Enter the number of elements in the array : ");
scanf("%d", &n);
printf("n Enter the elements of the array: ");
{
scanf("%d", &arr[i]);
}
printf("n The sorted array is: n");
printf(" %dt", arr[i]);
getch();
}
void merge(int arr[], int beg, int mid, int end)
{
while((i<=mid) && (j<=end))
{
if(arr[i] < arr[j])
{
}
else
{
}
}
if(i>mid)
{
while(j<=end)
{
MERGE_SORT(ARR, BEG, END)
Step 1: IF BEG < END
SET MID = (BEG + END)/2
CALL MERGE_SORT (ARR, BEG, MID)
CALL MERGE_SORT (ARR, MID + 1, END)
MERGE (ARR, BEG, MID, END)
[END OF IF]
Step 2: END
Figure 14.9 Algorithm for merge sort

23. 446 Data Structures Using C
}
}
else
{
while(i<=mid)
{
}
}
}
void merge_sort(int arr[], int beg, int end)
{
int mid;
if(beg<end)
{
merge_sort(arr, beg, mid);
merge(arr, beg, mid, end);
}
}
14.11 QUICK SORT
Quick sort is a widely used sorting algorithm developed by C. A. R. Hoare that makes O(n log n)
comparisons in the average case to sort an array of n elements. However, in the worst case, it has
a quadratic running time given as O(n ). Basically, the quick sort algorithm is faster than other O(n
log n)
quadratic time. Quick sort is also known as partition exchange sort.
Like merge sort, this algorithm works by using a divide-and-conquer strategy to divide a single
unsorted array into two smaller sub-arrays.
The quick sort algorithm works as follows:
1. Select an element pivot from the array elements.
2. Rearrange the elements in the array in such a way that all elements that are less than the pivot
appear before the pivot and all elements greater than the pivot element come after it (equal
This is called the partition operation.
3. Recursively sort the two sub-arrays thus obtained. (One with sub-list of values smaller than
that of the pivot element and the other having higher value elements.)
Like merge sort, the base case
because in that case the array is already sorted. After each iteration, one element (pivot) is always
element in the array as pivot.)

24. Searching and Sorting 447
Technique
loc
right
loc = = right = n
loc
a[loc] a[right]
right loc
right = loc
a[loc] > a[right]
loc = right
loc
a[loc]
loc
= loc
a[loc] <
loc =
Example 14.6
27 1 36 18 25 45
27 1 36 18 25 45
loc
left
right
We choose the first element as the pivot.
Set , , and .loc = left = right = 5
27 1 36 18 25 45
loc
left
right
Scan from right to left. Since
, decrease the value of .
a[loc]
< a[right] right
25 1 36 18 27 45
left right
loc
Since , interchange
the two values and set .
a[loc] > a[right]
loc = right
25 1 36 18 27 45
left right
loc
Start scanning from left to right. Since a[loc]
> a[left], increment the value of .left
25 1 27 18 36 45
left
loc
right
Since , interchangea[loc] < a[left]
the values and set .loc = left
25 1 27 18 36 45
left
loc
right
Scan from right to left. Since
, decrement the value of .
a[loc]
< a[right] right

25. 448 Data Structures Using C
25 1 18 27 36 45
left right
loc
Since , interchangea[loc] > a[right]
the two values and set .loc = right
25 1 18 27 36 45
right
loc
left
Start scanning from left to right. Since
, increment the value of .
a[loc]
> a[left] left
= loc
Partition
PARTITION (ARR, BEG, END, LOC)
Step 1: [INITIALIZE] SET LEFT = BEG, RIGHT = END, LOC = BEG, FLAG =
Step 2: Repeat Steps 3 to 6 while FLAG =
Step 3: Repeat while ARR[LOC] <= ARR[RIGHT] AND LOC != RIGHT
SET RIGHT = RIGHT - 1
[END OF LOOP]
Step 4: IF LOC = RIGHT
SET FLAG = 1
ELSE IF ARR[LOC] > ARR[RIGHT]
SWAP ARR[LOC] with ARR[RIGHT]
SET LOC = RIGHT
[END OF IF]
Step 5: IF FLAG =
Repeat while ARR[LOC] >= ARR[LEFT] AND LOC != LEFT
SET LEFT = LEFT + 1
[END OF LOOP]
Step 6: IF LOC = LEFT
SET FLAG = 1
ELSE IF ARR[LOC] < ARR[LEFT]
SWAP ARR[LOC] with ARR[LEFT]
SET LOC = LEFT
[END OF IF]
[END OF IF]
Step 7: [END OF LOOP]
Step 8: END
QUICK_SORT (ARR, BEG, END)
Step 1: IF (BEG < END)
CALL PARTITION (ARR, BEG, END, LOC)
CALL QUICKSORT(ARR, BEG, LOC - 1)
CALL QUICKSORT(ARR, LOC + 1, END)
[END OF IF]
Step 2: END
Figure 14.10

26. Searching and Sorting 449
Complexity of Quick Sort
In the average case, the running time of quick sort can be given as O(n log n). The partitioning of
the array which simply loops over the elements of the array once uses O(n) time.
In the best case, every time we partition the array, we divide the list into two nearly equal
pieces. That is, the recursive call processes the sub-array of half the size. At the most, only log n
nested calls can be made before we reach a sub-array of size 1. It means the depth of the call tree
is O(log n). And because at each level, there can only be O(n), the resultant time is given as O(n
log n) time.
O(n2
). The worst case occurs when the array is already sorted
(either in ascending or descending order) and the left-most element is chosen as the pivot.
However, many implementations randomly choose the pivot element. The randomized version
of the quick sort algorithm always has an algorithmic complexity of O(n log n).
Pros and Cons of Quick Sort
It is faster than other algorithms such as bubble sort, selection sort, and insertion sort. Quick sort
complex and massively recursive.
Programming ExamPlE
9. Write a program to implement quick sort algorithm.
#include <stdio.h>
#include <conio.h>
int partition(int a[], int beg, int end);
void quick_sort(int a[], int beg, int end);
void main()
{
printf("n Enter the number of elements in the array: ");
scanf("%d", &n);
printf("n Enter the elements of the array: ");
{
scanf("%d", &arr[i]);
}
printf("n The sorted array is: n");
printf(" %dt", arr[i]);
getch();
}
int partition(int a[], int beg, int end)
{
loc = left = beg;
right = end;
{

27. 450 Data Structures Using C
if(loc==right)
else if(a[loc]>a[right])
{
temp = a[loc];
a[loc] = a[right];
a[right] = temp;
loc = right;
}
{
if(loc==left)
else if(a[loc] <a[left])
{
temp = a[loc];
a[loc] = a[left];
a[left] = temp;
loc = left;
}
}
}
return loc;
}
void quick_sort(int a[], int beg, int end)
{
int loc;
if(beg<end)
{
loc = partition(a, beg, end);
}
}
14.12 RADIX SORT
Radix sort is a linear sorting algorithm for integers and uses the concept of sorting names in
alphabetical order. When we have a list of sorted names, the radix is 26 (or 26 buckets) because
there are 26 letters in the English alphabet. So radix sort is also known as bucket sort. Observe
contains the names with B, and so on.
During the second pass, names are grouped according to the second letter. After the second
nth
pass, where n
is the length of the name with maximum number of letters.
from the second bucket, and so on.
When radix sort is used on integers, sorting is done on each of the digits in the number. The
sorting the numbers, we have ten buckets, each for one digit (0, 1, 2, …, 9) and the number of
passes will depend on the length of the number having maximum number of digts.

28. Searching and Sorting 451
Algorithm for RadixSort (ARR, N)
Step 1: Find the largest number in ARR as LARGE
Step 2: [INITIALIZE] SET NOP = Number of digits in LARGE
Step 3: SET PASS =
Step 4: Repeat Step 5 while PASS <= NOP-1
Step 5: SET I = and INITIALIZE buckets
Step 6: Repeat Steps 7 to 9 while I<N-1
Step 7: SET DIGIT = digit at PASSth place in A[I]
Step 8: Add A[I] to the bucket numbered DIGIT
Step 9: INCEREMENT bucket count for bucket numbered DIGIT
[END OF LOOP]
Step 1 : Collect the numbers in the bucket
[END OF LOOP]
Step 11: END
Figure 14.11
Example 14.7
Number 0 1 2 3 4 5 6 7 8 9
Number 0 1 2 3 4 5 6 7 8 9

29. 452 Data Structures Using C
In the third pass, the numbers are sorted according to the digit at the hundreds place. The
buckets are pictured upside down.
Number 0 1 2 3 4 5 6 7 8 9
911 911
123 123
924 924
345 345
654 654
555 555
567 567
472 472
After the third pass, the list can be given as
123, 345, 472, 555, 567, 654, 808, 911, 924.
Complexity of Radix Sort
To calculate the complexity of radix sort algorithm, assume that there are n numbers that have to be
sorted and k is the number of digits in the largest number. In this case, the radix sort algorithm is
called a total of k times. The inner loop is executed n times. Hence, the entire radix sort algorithm
takes O(kn)
of numbers), then the algorithm runs in O(n) asymptotic time.
Pros and Cons of Radix Sort
Radix sort is a very simple algorithm. When programmed properly, radix sort is one of the fastest
sorting algorithms for numbers or strings of letters.
But there are certain trade-offs for radix sort that can make it less preferable as compared to
other sorting algorithms. Radix sort takes more space than other sorting algorithms. Besides the
array of numbers, we need 10 buckets to sort numbers, 26 buckets to sort strings containing only
characters, and at least 40 buckets to sort a string containing alphanumeric characters.
Another drawback of radix sort is that the algorithm is dependent on digits or letters. This
type, the algorithm has to be rewritten. Even if the sorting order changes, the algorithm has to
be rewritten. Thus, radix sort takes more time to write and writing a general purpose radix sort
algorithm that can handle all kinds of data is not a trivial task.
Radix sort is a good choice for many programs that need a fast sort, but there are faster sorting
algorithms available. This is the main reason why radix sort is not as widely used as other sorting
algorithms.
Programming ExamPlE
10. Write a program to implement radix sort algorithm.
##include <stdio.h>
#include <conio.h>
int largest(int arr[], int n);
void radix_sort(int arr[], int n);
void main()

30. Searching and Sorting 453
{
printf("n Enter the number of elements in the array: ");
scanf("%d", &n);
printf("n Enter the elements of the array: ");
{
scanf("%d", &arr[i]);
}
radix_sort(arr, n);
printf("n The sorted array is: n");
printf(" %dt", arr[i]);
getch();
}
int largest(int arr[], int n)
{
{
if(arr[i]>large)
large = arr[i];
}
return large;
}
void radix_sort(int arr[], int n)
{
large = largest(arr, n);
{
}
{
{
// sort the numbers according to the digit at passth place
bucket[remainder][bucket_count[remainder]] = arr[i];
}
// collect the numbers after PASS pass
{
{
arr[i] = bucket[k][j];
}
}
}
}
}

31. 454 Data Structures Using C
14.13 HEAP Sort
H
H
O(n log n)
n
∑ H
∑
H
Complexity of Heap Sort
insertion root deletion
H H H
m m H
g(n) to insert n in H
g(n) <= n log n
O(n log n)
H m
L reheaping
L H H O(log m)
log m L in H
Since n H n
n
O(n log n)
O(n log n)
n O(n log n)
Programming ExamPlE
11.
#include <stdio.h>
#include <conio.h>
HEAPSORT(ARR, N)
Step 1: [Build Heap H]
Repeat for I = to N-1
CALL Insert_Heap(ARR, N, ARR[I])
[END OF LOOP]
Step 2: (Repeatedly delete the root element)
Repeat while N>
CALL Delete_Heap(ARR, N, VAL)
SET N = N + 1
[END OF LOOP]
Step 3: END
Figure 14.12

32. Searching and Sorting 455
int main()
{
{
{
{
{
{
while(j<=n)
{

33. 456 Data Structures Using C
14.14 SHELL Sort
∑
∑
S
O(n )
n
Technique
∑ Step 1:
∑ Step 2:
Example 14.8
Solution
Result:
Result:

34. Searching and Sorting 457
Result:
Result:
Arr
Shell_Sort(Arr, n)
Step 1: SET FLAG = 1, GAP_SIZE = N
Step 2: Repeat Steps 3 to 6 while FLAG = 1 OR GAP_SIZE > 1
Step 3: SET FLAG =
Step 4: SET GAP_SIZE = (GAP_SIZE + 1) / 2
Step 5: Repeat Step 6 for I = to I < (N - GAP_SIZE)
Step 6: IF Arr[I + GAP_SIZE] > Arr[I]
SWAP Arr[I + GAP_SIZE], Arr[I]
SET FLAG =
Step 7: END
Figure 14.13

35. 458 Data Structures Using C
Programming ExamPlE
12. Write a program to implement shell sort algorithm.
#include<stdio.h>
void main()
{
int arr[10]={-1};
{
{
{
}
}
}
}
}
14.15 TREE SORT
Atree sort is a sorting algorithm that sorts numbers by making use of the properties of binary search
binary search tree using the numbers
to be sorted and then does an in-order traversal so that the numbers are retrieved in a sorted order.
We will not discuss this topic in detail here because we assume that reader has already studied it
Complexity of Tree Sort Algorithm
Let us study the complexity of tree sort algorithm in all three cases.
Best Case Worst Case
O
n
O (n)
O (n) O
n O n )
O (n)
O n2
O n O n2

36. Searching and Sorting 459
Programming ExamPlE
13.
#include <stdio.h>
#include <conio.h>
#include <alloc.h>
struct tree
{
{
{
{
else
{
else
{
{

37. 460 Data Structures Using C
14.16 coMPArISon oF SortInG ALGorItHMS
14.17 EXtErnAL SortInG
Example 14.9
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Disk
Input 1
Input 2
Input N
Output
Disk
I/O buffers in RAM
Figure 14.14
Generalized External Merge Sort Algorithm
K, then K
K K K
, K
table 14.1
Algorithm Average case Worst case
2
)
2
)
2
.k)
2
)
2
)
2
)
2
)
–
2
n)
Merge sort
Heap sort
Quick sort 2
)

38. Searching and Sorting 461
∑
∑
Applications of External Sorting
Projection Join
Points to rEmEmbEr
∑
∑
∑
∑
∑
∑
∑
∑
∑
∑
Divide
n
Conquer
Combine means
each
n
O(n log n)
∑
∑
∑
∑
∑